I. Introduction

II. Study Design and Material

Notation:

\[ \begin{aligned} k:& \text{subject}, k = 1,...,180\\ j:& \text{sequence}, j=0,1,2\\ i:& \text{period}, i=1,2,3\\ h:& \text{week}, h=1,2,3,4\\ \pi:& \text{period effect}\\ \tau:& \text{treatment effect}\\ \alpha:& \text{grouped treatment effects}\\ \beta:& \text{demographic effects}\\ \lambda:& \text{sequence/carryover effect}\\ \gamma:& \text{week effect}\\ b:& \text{subject-specific intercepts}\\ \mu:& \text{fixed-effect intercept}\\ \varepsilon:& \text{residual error}\\ \end{aligned} \]

Primary Objectives: Adverse Event

Proportion of Adverse Events per period

Patient Distribution: 30 people in per treatment sequence Initial Observation: Most patients do not face adverse events over the course of 4 weeks in each of the consecutive periods.

add demographic data

demographic summary

Distributional Summary of Predictor variables in the Study
period1
period2
period3
Characteristic No
N = 164
Yes
N = 16
p-value No
N = 168
Yes
N = 12
p-value No
N = 163
Yes
N = 17
p-value
age 31 (25, 37) 42 (40, 44) <0.001 32 (25, 38) 40 (37, 44) <0.001 32 (25, 38) 38 (32, 42) 0.012
1 Median (Q1, Q3)
2 Wilcoxon rank sum test

GLMM

Model Selection

LRT Between Model w/ and w/o Interaction Between Tx and Period
term npar AIC BIC logLik minus2logL statistic df p.value
glmm2 12 278.67 330.17 -127.34 254.67 NA NA NA
glmm1 16 281.34 350.01 -124.67 249.34 5.33 4 0.25
LRT Between Model w/ and w/o Race, Gender and Carryover Effect
term npar AIC BIC logLik minus2logL statistic df p.value
glmm3 7 271.25 301.29 -128.62 257.25 NA NA NA
glmm2 12 278.67 330.17 -127.34 254.67 2.57 5 0.77

Model Result (with Carryover Effect)

Characteristic OR 95% CI p-value
(Intercept) 0.00 0.00, 0.00 <0.001
Treatment


    Pill A
    Gel B 1.20 0.55, 2.64 0.6
    Gel C 0.84 0.37, 1.92 0.7
Period


    period1
    period2 0.71 0.31, 1.64 0.4
    period3 1.13 0.52, 2.46 0.8
Treatment Sequence


    ABC or BAC
    CAB or ACB 0.61 0.23, 1.62 0.3
    BCA or CBA 0.84 0.35, 2.02 0.7
age 1.19 1.11, 1.27 <0.001
gender


    Female
    Male 0.90 0.43, 1.89 0.8
race


    black
    others 1.55 0.61, 3.95 0.4
    white 1.74 0.65, 4.68 0.3
Abbreviations: CI = Confidence Interval, OR = Odds Ratio

Final Model Result (without Carryover Effect)

Notation: \[ \begin{aligned} \widehat{logit}(P(AE_{ijk} = 1)) &= b_{k} + \mu +\tau_{i} +\pi_j + \beta{a_{k}},\\ &\quad i = 1, 2, 3,\ j = 1, 2, 3,\ k = (1, 2, \ldots, 180),\\ &\quad b_{k} \sim N(0, {{\sigma}^2}_b) \end{aligned} \]

Characteristic OR 95% CI p-value
(Intercept) 0.00 0.00, 0.00 <0.001
Treatment


    Pill A
    Gel B 1.18 0.54, 2.58 0.7
    Gel C 0.85 0.37, 1.94 0.7
Period


    period1
    period2 0.72 0.31, 1.65 0.4
    period3 1.11 0.51, 2.40 0.8
age 1.19 1.12, 1.27 <0.001
Abbreviations: CI = Confidence Interval, OR = Odds Ratio

AE Prediction

Model Diagnostics

Secondary Objectives

Table 1. Baseline Characteristics
Characteristic Overall
N = 180
1
ABC
N = 30
1
ACB
N = 30
1
BAC
N = 30
1
BCA
N = 30
1
CAB
N = 30
1
CBA
N = 30
1
p-value2
Age (years) 32 (8) [18, 45] 33 (8) [19, 44] 31 (7) [18, 42] 31 (7) [20, 44] 33 (8) [19, 45] 32 (7) [21, 44] 31 (7) [20, 44] 0.6
Race






0.9
    Black 64 (36%) 12 (40%) 13 (43%) 12 (40%) 10 (33%) 8 (27%) 9 (30%)
    White 52 (29%) 8 (27%) 9 (30%) 7 (23%) 7 (23%) 12 (40%) 9 (30%)
    Other 64 (36%) 10 (33%) 8 (27%) 11 (37%) 13 (43%) 10 (33%) 12 (40%)
Female 81 (45%) 13 (43%) 11 (37%) 16 (53%) 14 (47%) 12 (40%) 15 (50%) 0.8
1 Mean (SD) [Min, Max]; n (%)
2 Kruskal-Wallis rank sum test; Pearson’s Chi-squared test

preliminary assessment and comparison of systemic and local Pharmacokinetics (PK) of Pill A, Gel B, and Gel C and the correlation of PK with adherence measures and the occurrence of adverse events

Blood PK

\[ Y_{ik} = \mu + b_{k} + \pi_i + \tau_{i} + \lambda_{i} + \alpha_{ik} + \beta_{ik} + \varepsilon_{ik}, \quad b_{k} \sim N(0, \sigma_b^2), \quad \varepsilon_{ik} \sim N(0, \sigma^2) \]

## Linear mixed model fit by REML. t-tests use Satterthwaite's method [
## lmerModLmerTest]
## Formula: bloodVL_change ~ treatment + overall_adhere + overall_safety +  
##     period + (1 | ptid)
##    Data: merged
## 
## REML criterion at convergence: 4827.4
## 
## Scaled residuals: 
##     Min      1Q  Median      3Q     Max 
## -3.5936 -0.5795  0.0153  0.6127  3.0873 
## 
## Random effects:
##  Groups   Name        Variance Std.Dev.
##  ptid     (Intercept)  23.09    4.805  
##  Residual             445.60   21.109  
## Number of obs: 540, groups:  ptid, 180
## 
## Fixed effects:
##                                Estimate Std. Error       df t value
## (Intercept)                     73.4552     6.7058 419.7422  10.954
## treatmentGel B                  -6.7916     2.3555 417.1354  -2.883
## treatmentGel C                  -9.0880     2.2967 391.5595  -3.957
## overall_adhere                   0.9365     0.1882 269.4894   4.975
## overall_safetyNo Adverse Event  26.4624     3.4675 521.9424   7.632
## periodperiod2                   -2.3295     2.2271 356.0129  -1.046
## periodperiod3                   -1.5433     2.2252 355.4545  -0.694
##                                            Pr(>|t|)    
## (Intercept)                    < 0.0000000000000002 ***
## treatmentGel B                              0.00414 ** 
## treatmentGel C                    0.000090106833765 ***
## overall_adhere                    0.000001161748202 ***
## overall_safetyNo Adverse Event    0.000000000000111 ***
## periodperiod2                               0.29629    
## periodperiod3                               0.48842    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Correlation of Fixed Effects:
##             (Intr) trtmGB trtmGC ovrll_ ov_NAE prdpr2
## treatmntGlB -0.437                                   
## treatmntGlC -0.359  0.538                            
## overall_dhr -0.832  0.328  0.247                     
## ovrll_sfNAE -0.616  0.080  0.032  0.196              
## periodperd2 -0.129 -0.009 -0.006 -0.026 -0.038       
## periodperd3 -0.173  0.001  0.001  0.004  0.009  0.499
Comparison Beta 95% CI p-value
Intercept (Pill A, Period 1, Sequence 0, No Adherence or AE) 73.5 60.3, 86.6 <0.001
Treatment


    Gel B vs Pill A -6.8 -11.4, -2.2 0.004
    Gel C vs Pill A -9.1 -13.6, -4.6 <0.001
Additional Day of Adherence 0.9 0.6, 1.3 <0.001
Adverse Event


    No Adverse Event 26.5 19.7, 33.3 <0.001
Period


    Period 2 vs Period 1 -2.3 -6.7, 2.1 0.3
    Period 3 vs Period 1 -1.5 -5.9, 2.8 0.5
Abbreviation: CI = Confidence Interval
##  contrast       estimate   SE  df t.ratio p.value
##  Pill A - Gel B     6.79 2.36 417   2.882  0.0115
##  Pill A - Gel C     9.09 2.30 391   3.956  0.0003
##  Gel B - Gel C      2.30 2.24 359   1.027  0.5602
## 
## Results are averaged over the levels of: overall_safety, period 
## Degrees-of-freedom method: kenward-roger 
## P value adjustment: tukey method for comparing a family of 3 estimates
##  contrast          estimate   SE  df t.ratio p.value
##  period1 - period2    2.330 2.23 355   1.046  0.5484
##  period1 - period3    1.543 2.23 355   0.694  0.7674
##  period2 - period3   -0.786 2.23 356  -0.353  0.9337
## 
## Results are averaged over the levels of: treatment, overall_safety 
## Degrees-of-freedom method: kenward-roger 
## P value adjustment: tukey method for comparing a family of 3 estimates

Skin PK

Linear Model

## 
## Call:
## lm(formula = skinVL_change ~ treatment + overall_adhere + overall_safety + 
##     period, data = merged)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -112.388  -22.128    1.035   20.921  115.662 
## 
## Coefficients:
##                                Estimate Std. Error t value             Pr(>|t|)
## (Intercept)                    105.6003     9.9680  10.594 < 0.0000000000000002
## treatmentGel B                 -24.4131     3.6314  -6.723       0.000000000046
## treatmentGel C                 -36.9951     3.5493 -10.423 < 0.0000000000000002
## overall_adhere                   1.8154     0.2753   6.594       0.000000000103
## overall_safetyNo Adverse Event  55.7940     5.2166  10.696 < 0.0000000000000002
## periodperiod2                    0.8567     3.4535   0.248                0.804
## periodperiod3                    1.6225     3.4507   0.470                0.638
##                                   
## (Intercept)                    ***
## treatmentGel B                 ***
## treatmentGel C                 ***
## overall_adhere                 ***
## overall_safetyNo Adverse Event ***
## periodperiod2                     
## periodperiod3                     
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 32.74 on 533 degrees of freedom
## Multiple R-squared:  0.3581, Adjusted R-squared:  0.3508 
## F-statistic: 49.55 on 6 and 533 DF,  p-value: < 0.00000000000000022
Comparison Beta 95% CI p-value
Intercept (Pill A, Period 1, Sequence 0, No Adherence or AE) 105.6 86.0, 125.2 <0.001
Treatment


    Gel B vs Pill A -24.4 -31.5, -17.3 <0.001
    Gel C vs Pill A -37.0 -44.0, -30.0 <0.001
Additional Day of Adherence 1.8 1.3, 2.4 <0.001
Adverse Event


    No Adverse Event 55.8 45.5, 66.0 <0.001
Period


    Period 2 vs Period 1 0.9 -5.9, 7.6 0.8
    Period 3 vs Period 1 1.6 -5.2, 8.4 0.6
Abbreviation: CI = Confidence Interval
##  contrast       estimate   SE  df t.ratio p.value
##  Pill A - Gel B     24.4 3.63 533   6.723 <0.0001
##  Pill A - Gel C     37.0 3.55 533  10.423 <0.0001
##  Gel B - Gel C      12.6 3.47 533   3.631  0.0009
## 
## Results are averaged over the levels of: overall_safety, period 
## P value adjustment: tukey method for comparing a family of 3 estimates
##  contrast          estimate   SE  df t.ratio p.value
##  period1 - period2   -0.857 3.45 533  -0.248  0.9666
##  period1 - period3   -1.623 3.45 533  -0.470  0.8853
##  period2 - period3   -0.766 3.45 533  -0.222  0.9733
## 
## Results are averaged over the levels of: treatment, overall_safety 
## P value adjustment: tukey method for comparing a family of 3 estimates

Linear Mixed Model

## Linear mixed model fit by REML. t-tests use Satterthwaite's method [
## lmerModLmerTest]
## Formula: skinVL_change ~ treatment + overall_adhere + overall_safety +  
##     period + (1 | ptid)
##    Data: merged
## 
## REML criterion at convergence: 5269.4
## 
## Scaled residuals: 
##     Min      1Q  Median      3Q     Max 
## -3.4333 -0.6760  0.0316  0.6391  3.5333 
## 
## Random effects:
##  Groups   Name        Variance Std.Dev.
##  ptid     (Intercept)    0      0.00   
##  Residual             1072     32.74   
## Number of obs: 540, groups:  ptid, 180
## 
## Fixed effects:
##                                Estimate Std. Error       df t value
## (Intercept)                    105.6003     9.9680 533.0000  10.594
## treatmentGel B                 -24.4131     3.6314 533.0000  -6.723
## treatmentGel C                 -36.9951     3.5493 533.0000 -10.423
## overall_adhere                   1.8154     0.2753 533.0000   6.594
## overall_safetyNo Adverse Event  55.7940     5.2166 533.0000  10.696
## periodperiod2                    0.8567     3.4535 533.0000   0.248
## periodperiod3                    1.6225     3.4507 533.0000   0.470
##                                            Pr(>|t|)    
## (Intercept)                    < 0.0000000000000002 ***
## treatmentGel B                       0.000000000046 ***
## treatmentGel C                 < 0.0000000000000002 ***
## overall_adhere                       0.000000000103 ***
## overall_safetyNo Adverse Event < 0.0000000000000002 ***
## periodperiod2                                 0.804    
## periodperiod3                                 0.638    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Correlation of Fixed Effects:
##             (Intr) trtmGB trtmGC ovrll_ ov_NAE prdpr2
## treatmntGlB -0.429                                   
## treatmntGlC -0.354  0.534                            
## overall_dhr -0.826  0.311  0.234                     
## ovrll_sfNAE -0.627  0.079  0.033  0.206              
## periodperd2 -0.137 -0.008 -0.005 -0.025 -0.037       
## periodperd3 -0.180  0.001  0.001  0.004  0.009  0.499
## optimizer (nloptwrap) convergence code: 0 (OK)
## boundary (singular) fit: see help('isSingular')
Comparison Beta 95% CI p-value
Intercept (Pill A, Period 1, Sequence 0, No Adherence or AE) 105.6 86.0, 125.2 <0.001
Treatment


    Gel B vs Pill A -24.4 -31.5, -17.3 <0.001
    Gel C vs Pill A -37.0 -44.0, -30.0 <0.001
Additional Day of Adherence 1.8 1.3, 2.4 <0.001
Adverse Event


    No Adverse Event 55.8 45.5, 66.0 <0.001
Period


    Period 2 vs Period 1 0.9 -5.9, 7.6 0.8
    Period 3 vs Period 1 1.6 -5.2, 8.4 0.6
Abbreviation: CI = Confidence Interval
##  contrast       estimate   SE  df t.ratio p.value
##  Pill A - Gel B     24.4 3.63 412   6.721 <0.0001
##  Pill A - Gel C     37.0 3.55 388  10.422 <0.0001
##  Gel B - Gel C      12.6 3.47 359   3.631  0.0009
## 
## Results are averaged over the levels of: overall_safety, period 
## Degrees-of-freedom method: kenward-roger 
## P value adjustment: tukey method for comparing a family of 3 estimates
##  contrast          estimate   SE  df t.ratio p.value
##  period1 - period2   -0.857 3.45 356  -0.248  0.9666
##  period1 - period3   -1.623 3.45 355  -0.470  0.8853
##  period2 - period3   -0.766 3.45 356  -0.222  0.9733
## 
## Results are averaged over the levels of: treatment, overall_safety 
## Degrees-of-freedom method: kenward-roger 
## P value adjustment: tukey method for comparing a family of 3 estimates

Combined Table

Comparison
Blood Viral Load
Skin Viral Load
Beta 95% CI p-value Beta 95% CI p-value
Intercept (Pill A, Period 1, Sequence 0, No Adherence or AE) 73.5 60.3, 86.6 <0.001 105.6 86.0, 125.2 <0.001
Treatment





    Gel B vs Pill A -6.8 -11.4, -2.2 0.004 -24.4 -31.5, -17.3 <0.001
    Gel C vs Pill A -9.1 -13.6, -4.6 <0.001 -37.0 -44.0, -30.0 <0.001
Additional Day of Adherence 0.9 0.6, 1.3 <0.001 1.8 1.3, 2.4 <0.001
Adverse Event





    No Adverse Event 26.5 19.7, 33.3 <0.001 55.8 45.5, 66.0 <0.001
Period





    Period 2 vs Period 1 -2.3 -6.7, 2.1 0.3 0.9 -5.9, 7.6 0.8
    Period 3 vs Period 1 -1.5 -5.9, 2.8 0.5 1.6 -5.2, 8.4 0.6
Abbreviation: CI = Confidence Interval